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+\documentclass[final]{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage[scale=1.6]{beamerposter} % Use the beamerposter package for laying out the poster
+
+\usetheme{confposter} % Use the confposter theme supplied with this template
+
+\usepackage{color, bbm}
+\setbeamercolor{block title}{fg=dblue,bg=white} % Colors of the block titles
+\setbeamercolor{block body}{fg=black,bg=white} % Colors of the body of blocks
+\setbeamercolor{block alerted title}{fg=white,bg=dblue!70} % Colors of the highlighted block titles
+\setbeamercolor{block alerted body}{fg=black,bg=dblue!10} % Colors of the body of highlighted blocks
+% Many more colors are available for use in beamerthemeconfposter.sty
+
+%-----------------------------------------------------------
+% Define the column widths and overall poster size
+% To set effective sepwid, onecolwid and twocolwid values, first choose how many columns you want and how much separation you want between columns
+% In this template, the separation width chosen is 0.024 of the paper width and a 4-column layout
+% onecolwid should therefore be (1-(# of columns+1)*sepwid)/# of columns e.g. (1-(4+1)*0.024)/4 = 0.22
+% Set twocolwid to be (2*onecolwid)+sepwid = 0.464
+% Set threecolwid to be (3*onecolwid)+2*sepwid = 0.708
+
+\newlength{\sepwid}
+\newlength{\onecolwid}
+\newlength{\twocolwid}
+\newlength{\threecolwid}
+\setlength{\paperwidth}{48in} % A0 width: 46.8in
+\setlength{\paperheight}{40in} % A0 height: 33.1in
+\setlength{\sepwid}{0.024\paperwidth} % Separation width (white space) between columns
+\setlength{\onecolwid}{0.22\paperwidth} % Width of one column
+\setlength{\twocolwid}{0.464\paperwidth} % Width of two columns
+\setlength{\threecolwid}{0.708\paperwidth} % Width of three columns
+\setlength{\topmargin}{-1in} % Reduce the top margin size
+%-----------------------------------------------------------
+
+\usepackage{graphicx}  % Required for including images
+
+\usepackage{booktabs} % Top and bottom rules for tables
+
+
+
+%----------------------------------------------------------------------------------------
+%	TITLE SECTION 
+%----------------------------------------------------------------------------------------
+
+\title{Sparse Recovery for Graph Reconstruction } % Poster title
+
+\author{Eric Balkanski, Jean Pouget-Abadie} % Author(s)
+
+\institute{Harvard University} % Institution(s)
+%----------------------------------------------------------------------------------------
+\begin{document}
+\addtobeamertemplate{block end}{}{\vspace*{2ex}} % White space under blocks
+\addtobeamertemplate{block alerted end}{}{\vspace*{2ex}} % White space under highlighted (alert) blocks
+
+\setlength{\belowcaptionskip}{2ex} % White space under figures
+\setlength\belowdisplayshortskip{2ex} % White space under equations
+
+\begin{frame}[t] % The whole poster is enclosed in one beamer frame
+
+\begin{columns}[t] % The whole poster consists of three major columns, the second of which is split into two columns twice - the [t] option aligns each column's content to the top
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The first column
+
+%----------------------------------------------------------------------------------------
+%	INTODUCTION
+%----------------------------------------------------------------------------------------
+
+
+%\vspace{- 15.2 cm}
+%\begin{center}
+%{\includegraphics[height=7em]{logo.png}} % First university/lab logo on the left
+%\end{center}
+
+%\vspace{4.6 cm}
+
+\begin{block}{Problem Statement}
+\begin{center}
+\bf{How can we reconstruct a graph on which observed cascades spread?}
+\end{center} 
+\end{block}
+
+
+%{\bf Graph Reconstruction}:
+
+%\begin{itemize}
+%\item \{${\cal G}, \vec p$\}: directed graph, edge probabilities
+%\item $F$: cascade generating model
+%\item ${\cal M} := F\{{\cal G}, \vec p\}$: cascade
+%\end{itemize}
+
+%{\bf Objective}:
+%\begin{itemize}
+%\item Find algorithm which computes $F^{-1}({\cal M}) = \{{\cal G}, \vec p\}$ w.h.p., i.e. recovers graph from cascades.
+%\end{itemize}
+
+%{\bf Approach}
+%\begin{itemize}
+%\item Frame graph reconstruction as a {\it Sparse Recovery} problem for two cascade generating models.
+%\end{itemize}
+
+%Given a set of observed cascades, the \textbf{graph reconstruction problem} consists of finding the underlying graph on which these cascades spread. We assume that these cascades come from the classical \textbf{Independent Cascade Model} where at each time step, newly infected nodes infect each of their neighbor with some probability.
+
+%In previous work, this problem has been formulated in different ways, including a convex optimization and a maximum likelihood problem. However, there is no known algorithm for graph reconstruction with theoretical guarantees and with a reasonable required sample size.
+
+%We formulate a novel approach to this problem in which we use \textbf{Sparse Recovery} to find the edges in the unknown underlying network. Sparse Recovery is the problem of finding the sparsest vector $x$ such that $\mathbf{M x =b}$. In our case, for each node $i$, we wish to recover the vector $x = p_i$ where $p_{i_j}$ is the probability that node $j$ infects node $i$ if $j$ is active. To recover this vector, we are given $M$, where row $M_{t,k}$ indicates which nodes are infected at time $t$ in observed cascade $k$,  and $b$, where $b_{t+1,k}$ indicates if node $i$ is infected at time $t+1$ in cascade $k$. Since most nodes have a small number of neighbors in large networks, we can assume that these vectors are sparse. Sparse Recovery is a well studied problem which can be solved efficiently and with small error if $M$ satisfies certain properties. In this project, we empirically study to what extent $M$ satisfies the Restricted Isometry Property.
+
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+\begin{block}{Voter Model}
+
+\begin{figure}
+\centering
+\includegraphics[width=0.6\textwidth]{images/voter.png}
+\end{figure}
+
+
+
+\vspace{0.5 cm}
+{\bf Description}
+
+\vspace{0.5 cm}
+
+\begin{itemize}
+\item $\mathbb{P}$({\color{blue} blue} at $t=0) = p_{\text{init}}$
+\item $\mathbb{P}$({\color{blue} blue} at $t+1) =  \frac{\text{Number of {\color{blue}blue} neighbors}}{\text{Total number of neighbors}}$
+\end{itemize}
+
+\vspace{0.5 cm}
+
+{\bf Sparse Recovery Formulation}
+
+\vspace{0.5 cm}
+
+
+To recover the neighbors of $v_1$, observe which nodes are  {\color{red} red} (1) or {\color{blue} blue} (0) at time step $t$:
+\begin{align*}
+&v_2 \hspace{0.2 cm}  v_3 \hspace{0.2 cm} v_4 \hspace{0.2 cm} v_5 &\\
+\vspace{1 cm}
+M  = &  \left( \begin{array}{cccc}
+0 & 0 & 1 & 1 \\
+1 & 1 & 0 & 0 \\
+\end{array} \right)  &  \begin{array}{l} \hspace{ - 5cm} 
+\text{time step 0} \\
+ \hspace{ - 5cm}  \text{time step 1} \\
+\end{array}
+\end{align*}
+
+and which color $v_1$ is at time step $t+1$ due to $M$:
+
+\begin{align*}
+b_1  = &  \left( \begin{array}{c}
+1 \\
+1 \\
+\end{array} \right) &  \begin{array}{l} \hspace{ - 5cm} 
+\text{time step 1} \\
+ \hspace{ - 5cm}  \text{time step 2} \\
+ \end{array}
+\end{align*}
+
+Then ,
+
+\begin{equation}
+\boxed{M \vec x_1 = \vec b_1 + \epsilon \nonumber}
+\end{equation}
+
+where $(\vec x_{1})_j := \frac{\text{1}}{\text{deg}(i)} \cdot \left[\text{j parent of 1 in }{\cal G}\right] $
+
+
+\end{block}
+
+
+
+
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+
+
+
+
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+
+
+\end{column} % End of the first column
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The first column
+
+%----------------------------------------------------------------------------------------
+%	CONSTRAINT SATISFACTION - BACKTRACKING
+%----------------------------------------------------------------------------------------
+\begin{block}{Independent Cascades Model}
+\begin{figure}
+\centering
+\includegraphics[width=0.6\textwidth]{images/icc.png}
+\end{figure}
+
+\vspace{0.5 cm}
+
+{\bf Description}
+
+\vspace{0.5 cm}
+
+\begin{itemize}
+\item Three possible states:   {\color{blue} susceptible},   {\color{red} infected},  {\color{yellow} inactive }
+\item $\mathbb{P}$(infected at t=0)$=p_{\text{init}}$
+\item Infected node $j$ infects its susceptible neighbors $i$ with probability $p_{j,i}$ independently
+\end{itemize}
+
+\vspace{0.5 cm}
+
+{\bf Sparse Recovery Formulation}
+
+\vspace{0.5 cm}
+
+To recover the neighbors of $v_5$,observe which nodes are  {\color{red} red} (1), {\color{blue} blue} (0), or {\color{yellow} yellow} (0) at time step $t$:
+\begin{align*}
+&v_1 \hspace{0.2 cm}  v_2 \hspace{0.2 cm} v_3 \hspace{0.2 cm} v_4 \\
+\vspace{1 cm}
+M  = &  \left( \begin{array}{cccc}
+1 & 0 & 0 & 0 \\
+0 & 1 & 1 & 0 \\
+\end{array} \right)   \begin{array}{l} \hspace{ 1cm} 
+\text{time step 0} \\
+ \hspace{ 1cm}  \text{time step 1} \\
+ \end{array}
+\end{align*}
+
+and if $M$ caused $v_5$ to be infected at time step $t+1$:
+
+\begin{align*}
+b_5  = &  \left( \begin{array}{c}
+0 \\
+1 \\
+\end{array} \right)  \begin{array}{l} \hspace{ 1cm} 
+\text{time step 1} \\
+ \hspace{ 1cm}  \text{time step 2} \\
+ \end{array}
+\end{align*}
+
+
+Then,
+
+\begin{equation}
+\boxed{e^{M \vec \theta_5} = (1 - \vec b_5) + \epsilon} \nonumber
+\end{equation}
+
+where $(\vec \theta_5)_j := \log ( 1 - p_{j,5})  $
+
+\vspace{1 cm}
+
+
+This problem is a {\bf Noisy Sparse Recovery} problem, which has been studied extensively. Here, the vectors $\vec x_i$ are deg(i)-sparse. 
+
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+
+
+
+
+
+
+
+
+
+%----------------------------------------------------------------------------------------
+%	MIP
+%----------------------------------------------------------------------------------------
+
+% \begin{block}{RIP property}
+
+% %The Restricted Isometry Property (RIP) characterizes a quasi-orthonormality of the measurement matrix M on sparse vectors. 
+
+% For all k, we define $\delta_k$ as the best possible constant such that for all unit-normed ($l$2) and k-sparse vectors $x$
+
+% \begin{equation}
+% 1-\delta_k \leq \|Mx\|^2_2 \leq 1 + \delta_k
+% \end{equation}
+
+% In general, the smaller $\delta_k$ is, the better we can recover $k$-sparse vectors $x$!
+ 
+% \end{block}
+
+%----------------------------------------------------------------------------------------
+
+\end{column} 
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The first column within column 2 (column 2.1)
+
+
+%----------------------------------------------------------------------------------------
+
+
+\begin{block}{Algorithms}
+
+{\bf Voter Model}
+
+\begin{itemize}
+\item Solve for each node i:
+\begin{equation}
+\min_{\vec x_i} \|\vec x_i\|_1 + \lambda \|M \vec x_i - \vec b_i \|_2 \nonumber
+\end{equation}
+\end{itemize}
+
+{\bf Independent Cascade Model}
+
+\begin{itemize}
+\item Solve for each node i:
+\begin{equation}
+\min_{\vec \theta_i} \|\vec \theta_i\|_1 + \lambda \|e^{M \vec \theta_i} - (1 - \vec b_i) \|_2 \nonumber
+\end{equation}
+\end{itemize}
+
+\end{block}
+
+\begin{block}{Restricted Isometry Property (RIP)}
+{\bf Definition}
+\begin{itemize}
+\item Characterizes a quasi-orthonormality of M on sparse vectors. 
+
+\item The RIP constant $\delta_k$ is the best possible constant such that for all unit-normed ($l$2) and k-sparse vectors $x$:
+
+\begin{equation}
+1-\delta_k \leq \|Mx\|^2_2 \leq 1 + \delta_k \nonumber
+\end{equation}
+
+\item The smaller $\delta_k$ is, the better we can recover $k$-sparse vectors $x$.
+\end{itemize}
+
+
+
+
+\end{block}
+
+\begin{block}{Theoretical Guarantees}
+
+With small RIP constants $(\delta \leq 0.25)$ for $M$ and some assumption on the noise $\epsilon$:
+
+{\bf Theorem \cite{candes}}
+
+If node $i$ has degree $\Delta$  and $n_{\text{rows}}(M) \geq C_1 \mu \Delta \log n$, then, w.h.p.,
+
+$$\| \hat x - x^* \|_2 \leq C (1 + \log^{3/2}(n))\sqrt{\frac{\Delta \log n}{n_{\text{rows}}(M) }}$$
+
+
+\end{block}
+
+
+
+
+%----------------------------------------------------------------------------------------
+%	RESULTS
+%----------------------------------------------------------------------------------------
+
+\begin{block}{RIP Experiments}
+
+\begin{center}
+\begin{table}
+\begin{tabular}{c | c | c | c | c }
+& $c$ = 100, &$c$ = 1000,& $c$ = 100, &$c$ = 1000,\\
+&  $i$ = 0.1& $i$ = 0.1& $i$ = 0.05& $i$ = 0.05\\
+ \hline
+  $\delta_4$ & 0.54 & 0.37 &0.43&0.23 \\
+  \end{tabular} 
+  \caption{RIP constant for a small graph. Here, $c$ is the number of cascades and $i$ is $p_{\text{init}}$.}
+\end{table}
+\end{center}
+
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+
+
+\end{column} % End of the second column
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The third column
+
+%----------------------------------------------------------------------------------------
+%	IOVERALL COMPARISON
+%----------------------------------------------------------------------------------------
+
+%\vspace{- 14.2 cm}
+%\begin{center}
+%{\includegraphics[height=7em]{cmu_logo.png}} % First university/lab logo on the left
+%\end{center}
+
+%\vspace{4 cm}
+
+\begin{alertblock}{Experimental Results}
+
+
+
+
+\end{alertblock} 
+
+%----------------------------------------------------------------------------------------
+
+
+%----------------------------------------------------------------------------------------
+%	CONCLUSION
+%----------------------------------------------------------------------------------------
+
+\begin{block}{Conclusion}
+
+\begin{center}
+
+
+{\bf Graph reconstruction can naturally be expressed as Sparse Recovery. Understanding properties of $M$, for example RIP, leads to theoretical guarantees on the reconstruction.}
+
+\end{center}
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+%	REFERENCES
+%----------------------------------------------------------------------------------------
+
+\begin{block}{References}
+
+\begin{thebibliography}{42}
+
+\bibitem{candes}
+Candès, E., and Plan, Y.
+\newblock {\it A Probabilistic and RIPless Theory of Compressed Sensing}
+\newblock Information Theory, IEEE Transactions on, 57(11): 7235--7254,
+\newblock 2011.
+\end{thebibliography}
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+
+\end{column} % End of the third column
+
+\end{columns} % End of all the columns in the poster
+
+\end{frame} % End of the enclosing frame
+
+
+
+\end{document}